Binary homomorphisms and natural dualities (Q1602678)

One of the most important questions in the theory of natural dualities is: ``Which finite algebras are dualisable?'' In connection with this question, the authors investigate ways in which certain binary homomorphisms of a finite algebra can guarantee its dualisability (of particular interest are those binary homomorphisms which are lattice, flat-semilattice or group operations). They prove that a finite algebra which has a pair of lattice operations amongst its binary homomorphisms is dualisable; as an application of this result, they find that every finite unary algebra can be embedded into a dualisable algebra. Also they develop some general tools that are used to prove the dualisability of a large number of unary algebras (for example is proved that the endomorphisms of a finite cyclic group are the operations of a dualisable unary algebra).